### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 040, 28 pages      arXiv:1710.08672      https://doi.org/10.3842/SIGMA.2018.040

### $({\mathfrak{gl}}_M, {\mathfrak{gl}}_N)$-Dualities in Gaudin Models with Irregular Singularities

Benoît Vicedo a and Charles Young b
a) Department of Mathematics, University of York, York YO10 5DD, UK
b) School of Physics, Astronomy and Mathematics, University of Hertfordshire, College Lane, Hatfield AL10 9AB, UK

Received November 06, 2017, in final form April 27, 2018; Published online May 03, 2018

Abstract
We establish $({\mathfrak{gl}}_M, {\mathfrak{gl}}_N)$-dualities between quantum Gaudin models with irregular singularities. Specifically, for any $M, N \in {\mathbb Z}_{\geq 1}$ we consider two Gaudin models: the one associated with the Lie algebra ${\mathfrak{gl}}_M$ which has a double pole at infinity and $N$ poles, counting multiplicities, in the complex plane, and the same model but with the roles of $M$ and $N$ interchanged. Both models can be realized in terms of Weyl algebras, i.e., free bosons; we establish that, in this realization, the algebras of integrals of motion of the two models coincide. At the classical level we establish two further generalizations of the duality. First, we show that there is also a duality for realizations in terms of free fermions. Second, in the bosonic realization we consider the classical cyclotomic Gaudin model associated with the Lie algebra ${\mathfrak{gl}}_M$ and its diagram automorphism, with a double pole at infinity and $2N$ poles, counting multiplicities, in the complex plane. We prove that it is dual to a non-cyclotomic Gaudin model associated with the Lie algebra ${\mathfrak{sp}}_{2N}$, with a double pole at infinity and $M$ simple poles in the complex plane. In the special case $N=1$ we recover the well-known self-duality in the Neumann model.

Key words: Gaudin models; dualities; irregular singularities.

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